A 19.5 gram, 56.3 centimeter long steel guitar string is vibrating in its fourth harmonic.  It induces a standing wave in a nearby 48.6 centimeter metal pipe open at one end.  The standing sound wave in the pipe corresponds to the pipe's fundamental frequency.

 
What is the tension on the steel guitar 'string'?  (in Newtons)

Just figured this one out! Okay, so:

The frequency of the pipe(f_p), and of the string (f_s) are both the same, since frequency is source dependent.
f_p=f_s
Now, f_p=n(v_snd/4L_p)
where n is 1 since the wave corresponds to the pipe's fundamental (first) frequency, v_snd is the speed of sound and L_p is the length of the pipe. For you, this value should come to: f_p=176.4403292...Hz, which is also f_s.

f_s=n(v_str/2L_s)
This time, we're looking for v_str (speed that the string vibrates). Rearranging gives us:
v_str=[2(L_s)(f_s)]/n
where n=4 since the string vibrates in its fourth harmonic, so your v_str=49.66795267...m/s

Finally, using the formula
v_str=√[T/(m/L_s)] and rearranging gives you T=m*((v_str)^2)/L_s
So your tension would be: 85.44344174...N
Check my work just in case!

Oh, btw, did you figure out questions 4,5, or 7?

Sorry for the late reply. Sadly, I didn't any of the ones you mentioned. Did you get #3?

For #4, it's F_tension=density x (lambda/T)^2

For #3, I put B. I dunno if it's the same for everyone's (since it's MC)

To find the tension in the steel guitar string, we need to use the formula for the fundamental frequency of a vibrating string:

f = (1/2L) * sqrt(T/μ)

Where:
f is the fundamental frequency,
L is the length of the string,
T is the tension in the string, and
μ is the linear mass density of the string.

Given that the string is vibrating in its fourth harmonic, we know that the length L of the string is equal to one-fourth of the wavelength of the fundamental frequency in the pipe, which is 48.6 cm. Therefore, L = (1/4) * 48.6 cm = 12.15 cm = 0.1215 m.

We are also given the linear mass density of the string is 19.5 g. Linear mass density (μ) is the mass per unit length, so we need to convert grams to kilograms: μ = 19.5 g/ 1000 = 0.0195 kg/m.

Since we are looking for the tension T, we can rearrange the formula and solve for T:

T = (f^2 * μ * 4L^2)

Now we need to know the fundamental frequency (f). Since the string is vibrating in its fourth harmonic, the fundamental frequency will be four times the frequency of the fourth harmonic. Let's denote the frequency of the fourth harmonic as 'f_4', so:

f_4 = 4 * f

To find 'f', we can use the formula for the frequency of a harmonic:

f = (n * v) / (2L)

Where:
n is the harmonic number,
v is the speed of sound in air, and
L is the length of the tube.

The pipe in this question is open at one end, so the fundamental frequency occurs when n = 1. Therefore, we can substitute n = 1 into the formula:

f = (v) / (2L)

The speed of sound in air at room temperature is approximately 343 m/s.

Let's calculate the fundamental frequency 'f' first:

f = 343 m/s / (2 * 0.486 m) = 353.91 Hz

Now, to find 'T', we substitute the values into the formula:

T = (f^2 * μ * 4L^2)
= (353.91^2 * 0.0195 kg/m * 4 * (0.1215 m)^2)

Simplifying this expression will give us the tension 'T' in Newtons.